Find the series expansion of $\frac{1}{\sqrt{1+5x}}$, up to and including the term in $x^2$. Sta

Question

Find the series expansion of $\frac{1}{\sqrt{1+5x}}$, up to and including the term in $x^2$. State the range of values of $x$ for which the expansion is valid.[3] Hence, use the substitution $x = \frac{1}{20}$ to obtain an approximation of $\sqrt{5}$, expressing your answer as a fraction.[2] Without any further calculation, explain whether using the substitution $x = -\frac{4}{25}$ gives a better approximation of $\sqrt{5}$ than the substitution used in part (b).[1]

Accepted Answer

(a) $1-\frac52x+\frac{75}{8}x^2+\cdots$, $-\frac15<x<\frac15$. (b) $\sqrt5\approx\frac{575}{256}$; the reciprocal route gives $\frac{256}{115}$. (c) No; the substitution is farther from zero.

2025 YIJC P1 Q3

2025 YIJC P1 Q3

Solution:

Solution locked